1 Intuitive Beliefs Jawwad Noor Boston University
2 Introduction • Economics: rational beliefs described by a prior probability + Bayesian updating • Psychology: – Beliefs are not monotone let alone additive (conjunction/disjunction fallacy) – Updating is not Bayesian (base-rate neglect, gambler’s fallacy, etc) – Intuitive judgements based on heuristics (Kahneman-Tversky 1974) • Field evidence for non-Bayesian updating – over/under updating: Stone (EI 2012) – gambler’s fallacy: Suetens et al (JEEA 2016) – Over-reaction: Debondt and Thaler (JF 1985) – Representativeness heuristic: Ahmed and Safdar (MS 2016)
3 Introduction • Behavior driven by intuition has economic relevance – incomplete deliberation due to limited time/cognitive resources – gaps filled by gut feeling • Classic reference on choice based on spontaneous feelings rather than deliberation: “[A] large proportion of our positive activities...can only be taken as the result of animal spirits—a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities.” Keynes (1936)
4 Introduction • This paper: formal theory of intuitive beliefs – Concepually, what is intuition? – How to model mathematically?
5 Introduction • This paper: formal theory of intuitive beliefs – Concepually, what is intuition? – How to model mathematically? • Inspiration from psychology and philosophy: associations • Activation of a mental image activates/inhibits another – e.g..... – Vocation, ethnicity, religion, etc may bring about the image of a stereotype – Involuntary and costless mental processing • Associations learnable, strength determined by frequency, salience, similarities, etc • Can be strengthened (reinforcement) or weakened (counter-conditioning or decay) • Intuitive beliefs driven by associations triggered by observation of information
6 Introduction • This paper: formal theory of intuitive beliefs – Concepually, what is intuition? – How to model mathematically? • Inspiration from cognitive science: neural networks • Elements of the model (note: not decision-theoretic): – network of associations, shaped by experience – nodes triggered by an observation generate output at connected nodes – the output of the network is what appears to us as intuition
7 Outline • Formal Model • Properties: Evidence, Bayesian intersection, Uniqueness • Axiomatization: Preview • Shaping the network • Conclusion
8 Primitives • A state of the world is a description of the relevant uncertainty: • Γ = { } = finite index set of elements of uncertainty • Ω = { } = abstract set of possible realizations of element ∈ Γ • Illustration: Agent is on a boat, lost in the open sea, and is uncertain about = health/survival = what kind of fish is swimming under the water
9 Primitives • A state of the world is a description of the relevant uncertainty: • Γ = { } = finite index set of elements of uncertainty • Ω = { } = abstract set of possible realizations of element ∈ Γ • Illustration: Agent is on a boat, lost in the open sea Γ = { } Ω = { } Ω = { }
10 Primitives • State space: Ω = Q ∈ Γ Ω , generic state = ( ) • Example Ω = { ( ) ( ) } • Large complicated state spaces are not a challenge • Construction and operation of networks is cognitively costless
11 Primitives • Σ = { Ω x y z } ⊂ 2 Ω = "elementary events" of element • Example: health/fish in the Yellow Sea... y ⊂ Ω = { } y ⊂ Ω = { } • Event space given by Σ = Q ∈ Γ Σ , generic event x = ( x x x ) • Identify Yellow Sea with y = ( y y ) ⊂ Ω × Ω • Product structure is convenient: each elementary event is a node in a network
12 Primitives • A belief ( ·| z ) conditional on z ∈ Σ over the space ( Ω Σ ) : – set function : Σ → [0 1] – assigns 1 to the full space – 0 to any event with a null elementary event – Satisfies ( x | z ) = ( x ∩ z | z ) – Not necessarily additive or monotone (conjunction/disjunction fallacies) • ( ·| Ω ) is the prior, ( ·| z ) for Ω 6 = ∈ Σ is posterior • Primitive: collection of conditional beliefs p = { ( ·| z ) : z ∈ Σ and ( z | Ω ) 0 }
13 Model • Each elementary event is a node of a network • {shark} • z • y • {death} • z • y
14 Model • Evaluate ( | y y ) • {shark} • z • y • {death} • z • y
15 Model • Evaluate ( | y y ) • {shark} • z % % • y − → − → − → − → − → • {death} • z • y ( | y ) ∈ R + ∪ { ∞ } ( | y ) ∈ R + ∪ { ∞ }
16 Model • Evaluate ( | y y ) • {shark} • z % &- % &- • y − → − → − → − → − → • {death} • z • y ( | y ) ( | ) × ( | y ) ( | y ) ( | ) × ( | y )
17 Model • Evaluate ( | y y ) ↑ • z • {shark} % &- % &- • y − → − → − → − → − → • {death} − → • z • y ( | y ) ( | ) × ( | y ) ( | y ) ( | ) × ( | y )
18 Model • Evaluate ( | y y ) ∙ • z • {shark} % ↑ &- % ↑ &- ⇒ • y − → − → ↑ − → − → • {death} ↑ % ↑ % • z • y ( | y ) ( | ) × ( | y ) ( | y ) ( | ) × ( | y ) ( | y ) ( | ) × ( | y ) ( | y ) ( | ) × ( | y )
19 Model • Expression for beliefs: Based on axioms and on tractability ( | y y ) = ⎡ ⎤ ( | y ) ( | ) × ( | y ) ⎢ ⎥ ( | y ) ( | ) × ( | y ) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ( | y ) ( | ) × ( | y ) ( | y ) ( | ) × ( | y )
20 Model • Expression for beliefs: Based on axioms and on tractability ( | y y ) = ⎡ ⎤ ( | y ) ( | ) × ( | y ) ⎢ ⎥ ( | y ) ( | ) × ( | y ) ⎢ ⎥ ⎢ ⎥ exp − 1 ⎢ ⎥ ⎣ ⎦ ( | y ) ( | ) × ( | y ) ( | y ) ( | ) × ( | y ) • Inverse of exp so that ∈ [0 1]
21 Model • Expression for beliefs: Based on axioms and on tractability ( | y y ) = ⎡ ⎤ ( | ) − 1 × ( | y ) − 1 ( | y ) − 1 ( | ) − 1 × ( | y ) − 1 ⎢ ⎥ ( | y ) − 1 ⎢ ⎥ ⎢ ⎥ exp − 1 ⎢ ⎥ ( | ) − 1 × ( | y ) − 1 ⎣ ⎦ ( | y ) − 1 ( | ) − 1 × ( | y ) − 1 ( | y ) − 1 • Inverse of exp so that ∈ [0 1] • Inverse of signals so that is increasing in signals
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